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6.3Separation of Variables and the Logistic Equation

Objective:

Solve differential equations that can be solved by separation of variables.

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Separation of VariablesAn equation of the form

is said to be separable and can be solved using separation of variables.

Original DE Rewrite

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Example 1: Find the general solution of

𝑥2+3 𝑦𝑑𝑦𝑑𝑥

=0

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Example 2: Find the general solution of

sin 𝑥 𝑦 ′=cos 𝑥

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Example 3: Find the general solution of

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Homogeneous Functions

is a homogenous function of degree if

(1)

(2)

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Homogeneous Differential Equations

A homogeneous differential equation is an equation of the form

where and are homogeneous functions of the same degree.

(1)

(2)

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Change of Variables for Homogenous Equations

If is homogenous, then it can be transformed into a DE whose variables are separable by the substitution

where is a differentiable function of .

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Example 1

(𝑥2− 𝑦2 )𝑑𝑥+3 𝑥𝑦 𝑑𝑦=0

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Example 2

𝑦 ′=𝑥3+𝑦3

𝑥 𝑦❑2

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Example 3

𝑦 ′=𝑥2+𝑦 2

2 𝑥𝑦

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Example 4

𝑦 ′=3𝑥+2 𝑦

𝑥

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Orthogonal Trajectories

Common problem in electrostatics, thermodynamics, and hydrodynamics

Involves finding a family of curves, each of which is orthogonal to all members of a given family of curves.

Ex. and Two such families are said to be mutually

orthogonal, and each curve in one family is called an orthogonal trajectory of the other family.

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Example 5

Describe the orthogonal trajectories for the family of curves given by

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Logistic Differential Equation

Exponential growth is unlimited, but when describing a population, there often exists some upper limit L past which growth cannot occur. This upper limit L is called the carrying capacity, which is the maximum population y(t) that can be sustained or supported as time t increases.

A model that is often used to describe this type of growth is the logistic differential equation

where k and L are positive constants.

From the equation, you can see that if y is between 0 and the carrying capacity L, then dy/dt > 0, and the population increases.

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Logistic Differential Equation

If y is greater than L, then dy/dt < 0, and the population decreases. The graph of the function y is called the logistic curve.

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Example 6 – Deriving the General Solution

Solve the logistic differential equation

Solution: